Step 1

Given function is \(\displaystyle{z}={x}^{{{2}}}{e}^{{{2}{y}}}\).

Partial derivative means derivative of function with respect to one variables keeping other variables as constant.

Finding partial derivative of the function with respect to x keeping y as constant.

\(\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({x}^{{{2}}}{e}^{{{2}{y}}}\right)}\)

\(\displaystyle={e}^{{{2}{y}}}\cdot{\frac{{\partial}}{{\partial{x}}}}{\left({x}^{{{2}}}\right)}\)

\(\displaystyle={e}^{{{2}{y}}}\cdot{2}{x}\)

\(\displaystyle={2}{{x}_{{{e}}}^{{{2}{y}}}}\)

Therefore, \(\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={2}{{x}_{{{e}}}^{{{2}{y}}}}\).

Step 2

Now, finding partial derivative with respect to y keeping other variable as constant.

\(\displaystyle{\frac{{\partial{z}}}{{\partial{y}}}}={\frac{{\partial}}{{\partial{y}}}}{\left({x}^{{{2}}}{e}^{{{2}{y}}}\right)}\)

\(\displaystyle={x}^{{{2}}}\cdot{2}{e}^{{{2}{y}}}\)

\(\displaystyle={2}{x}^{{{2}}}{e}^{{{2}{y}}}\)

Hence, \(\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={2}{x}{e}^{{{2}{y}}}{\quad\text{and}\quad}{\frac{{\partial{z}}}{{\partial{y}}}}={2}{x}{e}^{{{2}{y}}}\).

Given function is \(\displaystyle{z}={x}^{{{2}}}{e}^{{{2}{y}}}\).

Partial derivative means derivative of function with respect to one variables keeping other variables as constant.

Finding partial derivative of the function with respect to x keeping y as constant.

\(\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({x}^{{{2}}}{e}^{{{2}{y}}}\right)}\)

\(\displaystyle={e}^{{{2}{y}}}\cdot{\frac{{\partial}}{{\partial{x}}}}{\left({x}^{{{2}}}\right)}\)

\(\displaystyle={e}^{{{2}{y}}}\cdot{2}{x}\)

\(\displaystyle={2}{{x}_{{{e}}}^{{{2}{y}}}}\)

Therefore, \(\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={2}{{x}_{{{e}}}^{{{2}{y}}}}\).

Step 2

Now, finding partial derivative with respect to y keeping other variable as constant.

\(\displaystyle{\frac{{\partial{z}}}{{\partial{y}}}}={\frac{{\partial}}{{\partial{y}}}}{\left({x}^{{{2}}}{e}^{{{2}{y}}}\right)}\)

\(\displaystyle={x}^{{{2}}}\cdot{2}{e}^{{{2}{y}}}\)

\(\displaystyle={2}{x}^{{{2}}}{e}^{{{2}{y}}}\)

Hence, \(\displaystyle{\frac{{\partial{z}}}{{\partial{x}}}}={2}{x}{e}^{{{2}{y}}}{\quad\text{and}\quad}{\frac{{\partial{z}}}{{\partial{y}}}}={2}{x}{e}^{{{2}{y}}}\).